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I give a simple construction of the Coulomb branches of gauge theory in three and four dimensions, defined by H. Nakajima [Adv. Theor. Math. Phys. 20 (2016)] and A. Braverman, M. Finkelberg and H. Nakajima [Adv. Theor. Math. Phys. 22 (2018)] for a compact Lie group and a polarizable quaternionic representation . The manifolds are abelian group schemes over the bases of regular adjoint -orbits, respectively conjugacy classes, and is glued together over the base from two copies of shifted by a rational Lagrangian section , representing the Euler class of the index bundle of a polarization of . Extending the interpretation of as “classifying space” for topological 2D gauge theories, I characterize functions on as operators on the equivariant quantum cohomologies of , for compact symplectic -manifolds . The non-commutative version has a similar description in terms of the -class of .
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Constantin Teleman, The rôle of Coulomb branches in 2D gauge theory. J. Eur. Math. Soc. 23 (2021), no. 11, pp. 3497–3520